Calculus Review – some helpful hints

January 29, 2019 / Jonas Reitz / 0 Comments

Having trouble with the WeBWorK? First, don’t panic – it’s a lot to remember!! But do be prepared to put in some time re-learning stuff from Calculus I and II. I’ve picked out a few video resources for you that hit some of the most important techniques (I tried to find videos that were focussed on examples, rather than theory, since this is meant to be review).

Comments are welcome (just click the “leave a comment” button above).

Like a video? Leave comment and let me know.
Dislike a video (it wasn’t helpful/ it was confusing)? Let me know.
Need help with another topic (like the product rule, or equations of tangent lines, or something else)? Let me know.
Have a video or other resource to suggest? Let me know!

Derivatives: The Chain Rule (similar to Problem 4): This video is short and sweet, a single example using the chain rule with a logarithmic function.

Integrals: U-Substitution (similar to Problems 5 & 6):

This video has three examples – the first two are most similar to what you will see in WeBWorK (the last one is a little trickier – but could be useful in the future):

Integration by Parts (similar to Problems 7 & 8)

This video also has a few examples – the first two will be most useful for the WeBWorK:

Resources, Uncategorized

calculus chain rule differentiation integration integration by parts review substitution videos

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How to get/determine the Green's Function?
I want to determine the Green's function of $$t y^{\prime \prime}(t)+y^{\prime}(t)=t, \quad y(1)=y(e)=0.$$ I have even solved the ODE with the initial conditions, but I do not know how to determine the Green's function: We have a second-order linear ODE. Let $$\frac{dy(t)}{dt} = v(t),$$ which gives $$\frac{d^{2}y(t)}{dt^{2}} = \frac{dv(t)}{dt}.$$ Now, substitute $1 = \frac{d}{dt}(t):$ $$t […]
Power series solution to $x^2y''+y'+y=0$ around $x=0$
If an ODE of the form $$y''+p(x)y'+q(x)y=0$$ has $p(x)$ and $q(x)$ that are analytic at $x_0$ then we can suppose a power series solution as such: $$y(x)=\sum_{n\geq 0} a_n(x-x_0)^n$$ If instead they have a first order pole at $x_0$ we have to suppose this other one: $$y(x)=\sum_{n\geq 0} a_n(x-x_0)^{n+r}$$ But what about a power series […]
Contradicting solutions in Time and Fourier domains
Consider the following differential equations \begin{equation*} \partial_t X(t) = -i \alpha X(t) + \beta Y(t), \quad \partial_t Y(t) = -i \alpha Y(t) - \beta X(t).~~~~~ (1) \end{equation*} with $\alpha$, $\beta$ some complex constants (independent of $t$). Case-1 [Time domain]: The solution (with initial condition, say $X(0) = 1, Y(0) = 0$) reads \begin{equation*} X(t) = […]
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I am given the differential equation: $$ xyy' = y - 1 $$ I have tried to solve it by separating the variables, but I have only come this far: $$ \int 1 + \frac{1}{y-1} dy = \int \frac{1}{x} dx = y + \log|y-1| = \log{|x|} + C $$ I am struggeling on how to […]
Solution of an IVP
Given $$\frac{dy}{dx}+p(x)y=0$$ its general solution is $$y=c\cdot\exp\left(-\int p(x)dx\right)$$ where $c\in\mathbb{R}$. Let $y(x_{0})=y_{0}$. My question is: how am I supposed to use the initial condition here? What I did was: let $\mu(x)=-\int p(x)dx$. Then $$y_{0}=c\cdot\exp(\mu(x_{0}))\Rightarrow c=y_{0}\cdot\exp(-\mu(x_{0}))$$ If $\exp(-\mu(x_{0}))=C$, we have $$y=C\cdot y_{0}\cdot\exp\left(-\int p(x)dx\right)$$ PS: In my assignment, I have to choose the correct answer. There's an […]
How do I determine the stability of an equilibrium solution to a differential equation?
According to the notes issued out by my class, "an equilibrium solution $N (t) ≡ c$ to a differential equation (DE) is stable if $N(t)$ stays close to $c$ whenever $N(t)$ is a solution to the same DE such that $N(0)$ is sufficiently close to $c$." What does this mean? I'm unsure as to how […]
Do we have to set boundary at $0$?
Hi all I have a fundamental question about heat equation, so far all the heat equation set up I have experienced asasume boundary at a boundary at $x=0$, (assuming it's one-dimensional). I'm wondering what may happen if I say boundary is $x=1$ for example (the domain is $x>1, t>0$). Will that change the solution? I […]
How to convert a set of first order DEs to a single higher order DE
I'm wondering how to convert a set of first order DEs into just a single DE. I've been looking at this for a while and I can't figure it out. I understand the opposite direction, a single DE into a system of first order but I just can't figure this out. Specifically I'm looking at […]
Determining values of a parameter for which an initial value problem has a polynomial solution
Given the following IVP: \begin{cases} \ddot{y}-3t\dot{y}-ky=0\\ y(0)=1; \; \dot{y}(0)=0 \end{cases} I want to determine the possible values of $k$ in order to obtain a polynomial solution; if $p(t)=a_0+a_1t+ \dots + t^n$ would be a solution; then $a_0=1; a_1=0$; so I'd get $p(t)=1+ a_2t^2+ \dots + t^n$ and I also have that: $$(-3n-k)t^n=0 \iff k=-3n$$ But […]
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Struggling with this transfer function question

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The OpenLab is an open-source, digital platform designed to support teaching and learning at City Tech (New York City College of Technology), and to promote student and faculty engagement in the intellectual and social life of the college community.

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The OpenLab at City Tech:A place to learn, work, and share

The OpenLab is an open-source, digital platform designed to support teaching and learning at City Tech (New York City College of Technology), and to promote student and faculty engagement in the intellectual and social life of the college community.